Logarithmic Transform

Percentage Accurate: 41.5% → 99.3%

Time: 12.0s

Alternatives: 10

Speedup: 19.8×

• could not determine a ground truth

  1. c = 8.584509090018357e-79
  2. x = 4.186740123003922e+227
  3. y = 4.1682664174490517e-126

Specification

Math

\[\begin{array}{l} \\ c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow (E) x) 1.0) y)))))

Initial Program: 41.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow (E) x) 1.0) y)))))

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-27}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -2.5e-13) t_0 (if (<= y 9e-27) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -2.5e-13) {
		tmp = t_0;
	} else if (y <= 9e-27) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -2.5e-13) {
		tmp = t_0;
	} else if (y <= 9e-27) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -2.5e-13:
		tmp = t_0
	elif y <= 9e-27:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -2.5e-13)
		tmp = t_0;
	elseif (y <= 9e-27)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e-13], t$95$0, If[LessEqual[y, 9e-27], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999995e-13 or 9.0000000000000003e-27 < y

   1. Initial program 41.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 41.8

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 42.8

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 42.8

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 99.7

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   if -2.49999999999999995e-13 < y < 9.0000000000000003e-27

   1. Initial program 37.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 37.9

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 59.8

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 59.8

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 86.2

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 99.2

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-27}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -28000000000:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -28000000000.0)
   (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c)
   (if (<= y 1.06e-19)
     (* (* c (expm1 x)) y)
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -28000000000.0) {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	} else if (y <= 1.06e-19) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -28000000000.0)
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -28000000000.0], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.06e-19], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e10

   1. Initial program 52.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 52.5

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 52.5

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 52.5

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 99.7

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right) \cdot c \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right)} \cdot x\right)\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1\right) \cdot x\right)\right) \cdot c \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \cdot x\right)\right) \cdot c \]

      6. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \cdot x\right)\right) \cdot c \]

      7. lower-fma.f64 59.1

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \cdot x\right)\right) \cdot c \]

   7. Applied rewrites 59.1%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]

   if -2.8e10 < y < 1.06e-19

   1. Initial program 38.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 38.5

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 60.0

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 60.0

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 87.0

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 98.2

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   if 1.06e-19 < y

   1. Initial program 24.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(x \cdot \left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right)} \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right) \cdot x\right)} \cdot y\right) \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right) \cdot x\right)} \cdot y\right) \]

   5. Applied rewrites 52.1%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x, \frac{1}{6}\right), x, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x, \frac{1}{6}\right), x, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 52.1

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x, \frac{1}{6}\right), x, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x, \frac{1}{6}\right), x, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 99.7

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right)} \cdot c \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28000000000:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -28000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0
         (*
          (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y))
          c)))
   (if (<= y -28000000000.0)
     t_0
     (if (<= y 1.06e-19) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	double tmp;
	if (y <= -28000000000.0) {
		tmp = t_0;
	} else if (y <= 1.06e-19) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c)
	tmp = 0.0
	if (y <= -28000000000.0)
		tmp = t_0;
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -28000000000.0], t$95$0, If[LessEqual[y, 1.06e-19], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e10 or 1.06e-19 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 41.2

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 41.2

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 41.2

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 99.7

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right) \cdot c \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right)} \cdot x\right)\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x} + 1\right) \cdot x\right)\right) \cdot c \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right)} \cdot x\right)\right) \cdot c \]

      6. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x + \frac{1}{2}}, x, 1\right) \cdot x\right)\right) \cdot c \]

      7. lower-fma.f64 75.0

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \cdot x\right)\right) \cdot c \]

   7. Applied rewrites 75.0%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]

   if -2.8e10 < y < 1.06e-19

   1. Initial program 38.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 38.5

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 60.0

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 60.0

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 87.0

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 98.2

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28000000000:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c)))
   (if (<= y -1.1e+18) t_0 (if (<= y 1.06e-19) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	double tmp;
	if (y <= -1.1e+18) {
		tmp = t_0;
	} else if (y <= 1.06e-19) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c)
	tmp = 0.0
	if (y <= -1.1e+18)
		tmp = t_0;
	elseif (y <= 1.06e-19)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.1e+18], t$95$0, If[LessEqual[y, 1.06e-19], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e18 or 1.06e-19 < y

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 40.1

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 40.1

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 40.1

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 99.7

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)}\right) \cdot c \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) \cdot x\right)}\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) \cdot x\right)}\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} \cdot x\right)\right) \cdot c \]

      4. lower-fma.f64 73.7

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, x, 1\right)} \cdot x\right)\right) \cdot c \]

   7. Applied rewrites 73.7%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)}\right) \cdot c \]

   if -1.1e18 < y < 1.06e-19

   1. Initial program 39.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 39.3

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 60.6

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 60.6

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 87.2

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 97.7

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+181}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+181}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma y x 1.0)) c)))
   (if (<= y -4.2e+181) t_0 (if (<= y 2.4e+181) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -4.2e+181) {
		tmp = t_0;
	} else if (y <= 2.4e+181) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -4.2e+181)
		tmp = t_0;
	elseif (y <= 2.4e+181)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -4.2e+181], t$95$0, If[LessEqual[y, 2.4e+181], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.19999999999999995e181 or 2.40000000000000002e181 < y

   1. Initial program 45.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \color{blue}{\left(1 + x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto c \cdot \log \color{blue}{\left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + 1\right)} \]

      2. log-E N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot \color{blue}{1}\right) + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot \color{blue}{{1}^{2}}\right) + 1\right) \]

      4. log-E N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right) + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \log \left(\color{blue}{\left(x \cdot y\right) \cdot {\log \mathsf{E}\left(\right)}^{2}} + 1\right) \]

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {\color{blue}{1}}^{2} + 1\right) \]

      7. metadata-eval N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot \color{blue}{1} + 1\right) \]

      8. *-rgt-identity N/A

         \[\leadsto c \cdot \log \left(\color{blue}{x \cdot y} + 1\right) \]

      9. *-commutative N/A

         \[\leadsto c \cdot \log \left(\color{blue}{y \cdot x} + 1\right) \]

      10. lower-fma.f64 52.8

          \[\leadsto c \cdot \log \color{blue}{\left(\mathsf{fma}\left(y, x, 1\right)\right)} \]

   5. Applied rewrites 52.8%

      \[\leadsto c \cdot \log \color{blue}{\left(\mathsf{fma}\left(y, x, 1\right)\right)} \]

   if -4.19999999999999995e181 < y < 2.40000000000000002e181

   1. Initial program 38.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 38.6

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 53.1

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 53.1

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 91.2

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 85.6

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+181}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+181}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.85e+18) (* (* x y) c) (* (* c (expm1 x)) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.85e+18) {
		tmp = (x * y) * c;
	} else {
		tmp = (c * expm1(x)) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.85e+18) {
		tmp = (x * y) * c;
	} else {
		tmp = (c * Math.expm1(x)) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -1.85e+18:
		tmp = (x * y) * c
	else:
		tmp = (c * math.expm1(x)) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.85e+18)
		tmp = Float64(Float64(x * y) * c);
	else
		tmp = Float64(Float64(c * expm1(x)) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.85e+18], N[(N[(x * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e18

   1. Initial program 51.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. log-E N/A

         \[\leadsto c \cdot \left(x \cdot \left(y \cdot \color{blue}{1}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \left(x \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \]

      3. log-E N/A

         \[\leadsto c \cdot \left(x \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot {\log \mathsf{E}\left(\right)}^{2}\right)} \]

      5. log-E N/A

         \[\leadsto c \cdot \left(\left(x \cdot y\right) \cdot {\color{blue}{1}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto c \cdot \left(\left(x \cdot y\right) \cdot \color{blue}{1}\right) \]

      7. *-rgt-identity N/A

         \[\leadsto c \cdot \color{blue}{\left(x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

      9. lower-*.f64 42.5

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

   5. Applied rewrites 42.5%

      \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

   if -1.85e18 < y

   1. Initial program 35.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 35.9

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      5. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      6. lower-log1p.f64 52.5

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y}\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      9. lower-*.f64 52.5

         \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{x} - 1\right)}\right) \cdot c \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{{\mathsf{E}\left(\right)}^{x}} - 1\right)\right) \cdot c \]

      12. pow-to-exp N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{\log \mathsf{E}\left(\right) \cdot x}} - 1\right)\right) \cdot c \]

      13. lift-E.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\log \color{blue}{\mathsf{E}\left(\right)} \cdot x} - 1\right)\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{1} \cdot x} - 1\right)\right) \cdot c \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - 1\right)\right) \cdot c \]

      16. lower-expm1.f64 90.0

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \cdot c \]

   4. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(\left(e^{x} - 1\right) \cdot y\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(e^{x} - 1\right)\right) \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x} - 1\right) \cdot c\right)} \cdot y \]

      6. lower-expm1.f64 89.0

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(x\right)} \cdot c\right) \cdot y \]

   7. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot c\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.35e+23)
   (* (* c y) x)
   (* (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) c) x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.35e+23) {
		tmp = (c * y) * x;
	} else {
		tmp = ((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * c) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.35e+23)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * c) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.35e+23], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * c), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.3499999999999999e23

   1. Initial program 46.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)} \]

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot \color{blue}{1}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(1 \cdot y\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{y} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{c \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

      8. *-rgt-identity N/A

         \[\leadsto \left(c \cdot \color{blue}{\left(y \cdot 1\right)}\right) \cdot x \]

      9. metadata-eval N/A

         \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      10. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      11. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      12. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      13. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right) \cdot x} \]

      15. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      17. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      18. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      20. *-rgt-identity N/A

          \[\leadsto \left(c \cdot \color{blue}{y}\right) \cdot x \]

      21. lower-*.f64 67.4

          \[\leadsto \color{blue}{\left(c \cdot y\right)} \cdot x \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

   if 1.3499999999999999e23 < c

   1. Initial program 19.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(c \cdot \left(x \cdot \left(-3 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \left(2 \cdot \left({y}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(c \cdot \left(-1 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(2, y, -3\right), y\right), \left(y - y \cdot y\right) \cdot 0.5\right), x, c \cdot y\right) \cdot x} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(c + c \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot c\right) \cdot x\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.35e+23)
   (* (* c y) x)
   (* (* c x) (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.35e+23) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * y);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.35e+23)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.35e+23], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.3499999999999999e23

   1. Initial program 46.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)} \]

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot \color{blue}{1}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(1 \cdot y\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{y} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{c \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

      8. *-rgt-identity N/A

         \[\leadsto \left(c \cdot \color{blue}{\left(y \cdot 1\right)}\right) \cdot x \]

      9. metadata-eval N/A

         \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      10. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      11. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      12. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      13. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right) \cdot x} \]

      15. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      17. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      18. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      20. *-rgt-identity N/A

          \[\leadsto \left(c \cdot \color{blue}{y}\right) \cdot x \]

      21. lower-*.f64 67.4

          \[\leadsto \color{blue}{\left(c \cdot y\right)} \cdot x \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

   if 1.3499999999999999e23 < c

   1. Initial program 19.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(c \cdot \left(x \cdot \left(-3 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \left(2 \cdot \left({y}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(c \cdot \left(-1 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(2, y, -3\right), y\right), \left(y - y \cdot y\right) \cdot 0.5\right), x, c \cdot y\right) \cdot x} \]

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(c + c \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot c\right) \cdot x\right) \cdot \color{blue}{y} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.8%

       \[\leadsto \left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)\right) \cdot \left(c \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.35 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.65e+53) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.65e+53) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 1.65d+53) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.65e+53) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.65e+53:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.65e+53)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.65e+53)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.65e+53], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.6500000000000001e53

   1. Initial program 45.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)} \]

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot \color{blue}{1}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(1 \cdot y\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{y} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{c \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

      8. *-rgt-identity N/A

         \[\leadsto \left(c \cdot \color{blue}{\left(y \cdot 1\right)}\right) \cdot x \]

      9. metadata-eval N/A

         \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      10. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      11. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      12. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      13. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right) \cdot x} \]

      15. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      17. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      18. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      20. *-rgt-identity N/A

          \[\leadsto \left(c \cdot \color{blue}{y}\right) \cdot x \]

      21. lower-*.f64 67.4

          \[\leadsto \color{blue}{\left(c \cdot y\right)} \cdot x \]

   5. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

   if 1.6500000000000001e53 < c

   1. Initial program 19.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)} \]

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot \color{blue}{1}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(1 \cdot y\right)} \]

      4. *-lft-identity N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{y} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{c \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

      8. *-rgt-identity N/A

         \[\leadsto \left(c \cdot \color{blue}{\left(y \cdot 1\right)}\right) \cdot x \]

      9. metadata-eval N/A

         \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      10. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      11. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      12. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      13. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)\right) \cdot x \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right) \cdot x} \]

      15. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      16. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

      17. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

      18. log-E N/A

          \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

      20. *-rgt-identity N/A

          \[\leadsto \left(c \cdot \color{blue}{y}\right) \cdot x \]

      21. lower-*.f64 52.6

          \[\leadsto \color{blue}{\left(c \cdot y\right)} \cdot x \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.8%

      \[\leadsto \left(x \cdot c\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.65 \cdot 10^{+53}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.6% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \left(c \cdot y\right) \cdot x \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* (* c y) x))

C

double code(double c, double x, double y) {
	return (c * y) * x;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (c * y) * x
end function

Java

public static double code(double c, double x, double y) {
	return (c * y) * x;
}

Python

def code(c, x, y):
	return (c * y) * x

Julia

function code(c, x, y)
	return Float64(Float64(c * y) * x)
end

MATLAB

function tmp = code(c, x, y)
	tmp = (c * y) * x;
end

Wolfram

code[c_, x_, y_] := N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 39.6%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)} \]

   2. log-E N/A

      \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot \color{blue}{1}\right) \]

   3. *-commutative N/A

      \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(1 \cdot y\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{y} \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{c \cdot \left(x \cdot y\right)} \]

   6. *-commutative N/A

      \[\leadsto c \cdot \color{blue}{\left(y \cdot x\right)} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]

   8. *-rgt-identity N/A

      \[\leadsto \left(c \cdot \color{blue}{\left(y \cdot 1\right)}\right) \cdot x \]

   9. metadata-eval N/A

      \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

   10. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

   11. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

   12. metadata-eval N/A

       \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

   13. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right) \cdot x} \]

   15. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

   16. metadata-eval N/A

       \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{{1}^{2}}\right)\right) \cdot x \]

   17. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{2}\right)\right) \cdot x \]

   18. log-E N/A

       \[\leadsto \left(c \cdot \left(y \cdot {\color{blue}{1}}^{2}\right)\right) \cdot x \]

   19. metadata-eval N/A

       \[\leadsto \left(c \cdot \left(y \cdot \color{blue}{1}\right)\right) \cdot x \]

   20. *-rgt-identity N/A

       \[\leadsto \left(c \cdot \color{blue}{y}\right) \cdot x \]

   21. lower-*.f64 64.2

       \[\leadsto \color{blue}{\left(c \cdot y\right)} \cdot x \]

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]
6. Add Preprocessing

Developer Target 1: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (log1p (* (expm1 x) y))))

C

double code(double c, double x, double y) {
	return c * log1p((expm1(x) * y));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log1p((Math.expm1(x) * y));
}

Python

def code(c, x, y):
	return c * math.log1p((math.expm1(x) * y))

Julia

function code(c, x, y)
	return Float64(c * log1p(Float64(expm1(x) * y)))
end

Wolfram

code[c_, x_, y_] := N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024296 
(FPCore (c x y)
  :name "Logarithmic Transform"
  :precision binary64

  :alt
  (* c (log1p (* (expm1 x) y)))

  (* c (log (+ 1.0 (* (- (pow (E) x) 1.0) y)))))